

for all multi indices \(\alpha \in \mathbb {N}^2\)  such that \(|\alpha | \le q\) , where \(q\)  be the maximum of \(\operatorname{deg}(\Phi )+4\) , \(\operatorname{deg}(\nabla W\otimes \nabla W)+2\) , and \(\operatorname{deg}(N_1)\) . The equivalency of (REF ) and (REF ) can be established using Lemma REF . The second von Kármán equation (in a distributional form) is as given in (REF ) whose equivalence to the following local form can be established using [1]}, Identity REF , Identity REF , and Lemma REF :
\(D\Delta ^2 w - [\phi ,w]=f_1 + D\left( (1-\nu )\operatorname{div}\operatorname{div}{\lambda ^p} +\nu \Delta (\operatorname{tr}({\lambda ^p}))\right) ~ \text{in}~\Omega -\lbrace S\cup \lbrace O\rbrace \rbrace ,\\D \left\langle \llbracket \operatorname{div}\nabla \nabla w \rrbracket , \nu  \right\rangle = f_2 + D(1-\nu ) \langle \llbracket \operatorname{div}\lambda ^p \rrbracket , \nu  \rangle + D\nu \llbracket \nabla (\operatorname{tr}\lambda ^p) \rrbracket ~\text{on} ~S-\lbrace O\rbrace ,\\\llbracket \Delta w\rrbracket = 0 ~\text{on} ~S-\lbrace O\rbrace ,~\text{and}\\D W(\Delta ^2 v^\alpha )+\nabla \Phi \otimes \nabla W(\mathbb {A}\nabla \nabla v^\alpha )=F(v^\alpha )-D\left( (1-\nu )\Lambda ^p(\nabla \nabla v^\alpha ) +\nu (\operatorname{tr}(\Lambda ^p)(\Delta v^\alpha ))\right)~\text{at}~O,\) 
